THE SUMS OF CERTAIN SERIES CONTAINING HYPERBOLIC FUNCTIONS

F.P.SAYER University of Bristol, Bristol, England

and

1. liTRODUCTIOfy

In this paper we are concerned with the summation of a number of series. They are

± (-»"' y cothnr ^ »»*<»">% y j ^

^ / ^ s i n h r / j$ r4p~1 ' *£ (2r + V4p~1 ' % (2r + I)4***cosh (2r + 1)±

y (-1)rHr4p~3 y (-1)r(2r+1)4p~1

h , i n h n r ' h mhVr+Vf

~ | 2 4 p c o t h r | - c o t h 2 n r f

r=1 '

where p = 1, 2, 3, Certain of the above series have been extensively discussed in the past Results for particular values of/7 are given

by Ramanujan in [ 4 ] , while Phillips, Sandham and Watson in [3, 5, 6] have determined, by varying methods, sums for general p. The last series of the group, however, seems to have received less attention. It differs from the others in that it contains the inverse powers of 4p + /. Further, it is cioselv related to the Riemann Zeta function $(4p + 1). As this paper shows, the sums of the series, where they are not identically zero, satisfy recursive relations containing binomial coefficients.

Thus if we write

T4p-1 (~1)p(4p)! y (-1)r

^4P-122p-2 f - r 4 P - / g i n h n r

then

t ( V p2 ) V / " / n=1,2,

P=1

The recursive relations are themselves of interest and can be inverted. Their inversion, which leads to the sums of the various series, involves the Bernoulli and the lesser known Euler numbers.

All results are obtained by considering the Neumann problem for the rectangle. Although this problem is of an ele-mentary nature and is in fact discussed in both contemporary and established literature on Laplace's equation, a com-plete solution to it does not seem to be available. Kantorovich and Krylov in [2] proposed a method of solution but the suggested method contains, as we shall show, an error of principle. Once this error is removed the method can be applied to solve the problem. Initially, therefore, we state and solve the Neumann problem for the rectangle and then subsequently in Section 3 make appropriate use of the solution to obtain the various results.

215

216 THE SUMS OF CERTAIN SERIES CONTAINING HYPERBOLIC FUNCTIONS [OCT.

2. THE NEUMANN PROBLEM FOR THE RECTANGLE This problem requires the determination of a function </>(x,yJ satisfying

(2.1) ^xx + <t>yy - 0 for 0 < x < a, 0 < y < h

(2.2) <t>y(x,0) = f(x), <j)y(x,b) = g(x) for O < x < a

(2.3) 4>x(0,yJ = F(y), <px(a,y) = G(y) for 0 < y < b,

where fix), g(x), F(y) and G(y) are known functions and the subscripts* and y are used to denote partial differentiation. It is necessary for a solution that

(2.4) f |& ds = 0, c

where c is the boundary of the rectangle, d/dn denotes differentiation with respect to the outward normal to c and s refers to arc length. The condition (2.4) is equivalent to

a b (2.5) f (f-g)dx+ f (F-G)dy = 0.

0 0

We now briefly describe the method used by Kantorovich and Krylov in [2]. We put $ = U + V, where U and V are functionsofxandy. We choose the function U so that it satisfies (2.1)# (2.2) and Ux(0,y)= Ux(a,y) = £7 for 0<y < b, while Vsatisfies (2.1), (2.3) and Vy(x,0) = Vy(x,b) = OforO<x<a.

Thus, the original Neumann problem is replaced by two other Neumann problems, one for U and the other for V. It is evident that if we can find U and V we shall fulfill the conditions imposed on 0 by (2.1) to (2.3). By virtue of (2.4) the existence of U requires

a f (f-g)dx = 0.

0 Likewise, the existence of V requires

b f (F-G)dy = 0. o

However, given functions £& Fand £ satisfying (2.5), it does not necessarily follow that the integrals a b

f(f-g)dx and f(F-G)dy 0 0

are each zero, and therefore the functions U and V may not exist. Yet the difficulty is readily overcome. We write 0 = A(x2-y2) + U+K

where A is some constant to be found, while the functions (/and V each satisfy (2.1) and the further conditions: Ux{0.y) = Ux(a,y) - Vy(x,0) - Vy(x,b) - O

Uy(x,0) = f(x), Uy(x,b) = g(x)+2Av for O < x < a

Vx(0,y) = F(y), Vx(a,y) = G(y)-2Aa for 0 < y < b.

Using (2.4) we require for the existence of U and V

a a f | g(x) + 2Ab - fix) \dx = 0, i.e., 2abA = f (f-g)dx

0 0

and

1876] THE SUMS OF CERTAIN SERIES CONTAINING HYPERBOLIC FUNCTIONS 217

f { G(y)-2Aa - F(y) } dy = 0 or 2abA = f (G - F)dy. 0 0

Equation (2.5) shows that these two expressions for A are consistent Having found A, we can now follow the pro-cedure given in [2] to determine U and V. In fact, it can be verified directly that to within an arbitrary constant <£ is given by

\gr cosh 3K = fr cosh QL (b -y) \ (2.6) 0 = A(x* -y2) + 1/2fQy + 1/2F0x + V —* = —± 1 cos

r^ nrslnh ^ a r a

°° b i£?rcosh ^ ~Frcosh 2E fe-xjA + V - 1 - ^ — Jt i c o s ^ f ,

% nrsinh G5 * b

where r"r, #r fr = 0, 1, 2, •••) are the Fourier cosine coefficients for f(x) and #W, respectively, over the range 0 < x < a and Fr, Gr (r= 0, 1, 2, •••) are the Fourier cosine coefficients of F(y) and G(y) over 0 <y <b.

3. APPLICATION OF THE SOLUTION TO THE NEUMANN PROBLEM We puta = i& = 7rand define functions (p(x,y,4n), where n= 1,2,3, -., by

(3.1) 2$(x,y,4n) = (x + /y)4n + (x- iy)4n. It is readily verified that these functions satisfy (2.1). Further, using (2.2) and (2.3), we deduce for them that f(x) and Fly) are both identically zero. In addition

glx) = 2n { fr+ix)4n'1 + fr- ix)4n~f} and G(y) = 2n \ fr+iy)4n'1 + fr-iy)4n~1} . Thus, the Fourier coefficients fr and />are all zero, while gr = Gr = lr(n) (r= 1,2, —), where

(3.2) lr(n) = Re4-^ f [fr+ix)4n'1-t fr- ix)4n'1]eirxdx 0

using the result a

2abA = f (f-g)dx

o

we find that the constants vanishes and hence with the help of (2.6) we can write

(3.3) 4>(x,yAn) = < * „ + £ !,(„} )cosh/y cos^cosh/x cos/zl

the C4n (n = 1, 2, • •) being constants which have yet to be determined. Successive integration by parts of (3.2) leads to the result

(3.4) lr{n) = ( = ^ 7r4n'322n(4n)(4n -1) + % (4n - 1)(4n - 2)(4n - 3)1 Jin - 1) . :: r2 t4

In particular

so that putting n= 1 in (3.1) and (3.3) we find

E t i\r+l \ cos" ry cos rx + cosh rx cos ry § (~~V ' "Fl iShnr

!r(1) = h1)»l 4M

218 THE SUMS OF CERTAIN SERIES CONTAINING HYPERBOLIC FUNCTIONS [0CT.

Repeated application of (3.4) yields

(3.6) lrM - (-irl'41 * ̂ * ̂ * aj¥T-\ • [ r2 r6 r10

r4n-2 J where, for example, (3.7) a2(n) = (-1)n7T4n~322n4n(4n - / ; and more generally (3.8) a4p.2<n) = {-ir*+i*4n-4p+122n-2p+2(4p-2)! [4p

4n_2 ) , P =.1,2, - , n.

Using this last result, it follows a4p+2(n +U s (4n+4)(4n +3)(4n+2)(4n + 1)a4p-2(n)

and hence from (3.6) that

(3.9) fan +4)(4n + 3)(4n+2)(4n + 1) ^ = lr(n + i) + (~1)*1 a2(n + 1) .

We now proceed to find the constants C4n occurring in (3.3). We integrate Eq. (3.3) twice with respect to x and twice with respect to y. These integrations will introduce arbitrary functions of x and y. We have, therefore,

JihTl^&ilim*4l ^n(y^an(y)+yPnM+qnM

= c x2V2 _ V * / (n) I cosh ry cos rx + cosh rx cos ry f . 4n 4 Ld r rs sinh m

r=1

wherepn(x), qn(x), Pn(y) and Qn(y) are arbitrary functions which may depend on n. Noting the result contained in (3.9) we can write this equation in the alternative form

<j>(x,y,4n+4) = T l Ir(„ + 1) + (=1^1 ^ (n + ; A t cosh V c p i / * * cosh rx cos / y ] r=1 ^ J

+ (4n + 1)(4n+2)(4n+3)(4n+4) J xPn(y) + Qn(y) + yPn (x) + qn (x)- c4n ^ f \ -

This reduces with the help of (3.3) and (3.5) to

0 = -c4n+4 + ̂ p - (** - Bx*y2 +y*-cj

+ (4n + 1){4n+2M4n+3)(4n + 4) <xPn(y) +Qn(y) + ypn(x) + qn(x)-c4n x-^f-\ .

This is an identity. Hence equating to zero the coefficient of x2y2 we deduce with the aid of (3.7) / 1 \n~An 02n

(3-10) *» - 0ijkhr • Thus we have (3.11) (x+iy)4n + (x-iy)4n = 2c4n + 2 V lr(n) l£0.sh ry cos rx ^cosJLaf^o»f lJ

*-* rsinhnr r=l

where c4n is given by (3.10) and lr(n) by results (3.6) and (3.8). Putting* = y = 0in (3.11) and simplifying we obtain

(4n + 1)(4n + 2) ^ r s i n h n r | Z * 7t4p-i22p-2r4p-2 I ' *' '

1976] THE SUMS OF CERT AIM SERIES COWTAIMIWG HYPERBOLIC FUf\SCTIOWS 219

Thus if we write oo

•1)r

, P = 1,2, -sinh m

(3.12) T4P.1 = (-7)P(4p,! T ~ L

then it follows T4P-i satisfies the recursive relation

(us) /= £ (4»4;2| Upi n = lz..._

This is the first of our results. We now show how this recursive relation can be inverted to give F ^ / in terms of the Bernoulli numbers. To do this, we observe that (3.13) can be put in the alternative form

T 1 _ T~"% '4p-1

(4n+2)l 2-J (4p)!(4n+2~4p)!

Multiplying both sides of this equation by x and summing from n = 1 to <*> yields

1J (4n+2)l " 2-r LJ (4p)!(4n+2-4p)! ) ^ (4p)l [ ) £ i (4k + 2)!

After some manipulation we obtain

°° A (3.14) V 1 UD-1 77-iT = 7 - —r-^~ = 1+ ^cosechcurcosech/cur,

^ w ^ f4/?// cosh x - cos x 2 P=1

where 2a = 1 + i. Using the expansion of cosechx given in [1] Eq. (3.15) leads after some simplification to

T«~' ' ^ Z (-'ft**-'- D(24p-2c>-1 - D ( J ) BqB2p.q . 2 q=0

It should be noted that BQ is taken as - 1 while the Bernoulli numbers are defined here by

?V-j' i>""'*&-e —/ p=i

With the help of (3.12) we deduce

In a similar manner if we pu t * = y = 7rin (3.11) and define S4p~t by

oo

S4p-1 = (-1)P~1*1-4p2-2p+2(4p)! Y ^-^ " Ap-1

then n

(3.16) £ [*£2)s4p-t-'*>(*!>+3)

220 THE SUMS OF CERTAIN SERIES CONTAINING HYPERBOLIC FUNCTIONS [OCT.

S4P~i can also be expressed in terms of the Bernoulli numbers. By writing (3.16) in the form

V S4p~1 = I / —L l—\ L* (4n -4p + 2)!(4p)! 2 \ (4n)! (4n + 2)! \

and following a procedure similar to that for T^p-/ we find

J ^ x4p 2 > S4P-i * ,/ = - y - coth ax coth lax - /.

Since (see [1]!

we have

giving

2p xz*xhx= £ r - / / * V * jf^

p=0

«p-i-**f: <-vp+q {%) *qB2P~q q=0

n\i\ V c o t h nr = P4p-2_4p-1 TT* (-Dq+ Bg%2p-g [ } JL, 4f>-1 Z * jLi (2q)!(4p-2q)! '

r=1 i q=Q

We next put x = 0, y = -n in (3.11) and subtract from twice the result the expressions obtained by putting* = y = 0 and x = y = n. This leads to

(3.18) ^"U + l-n^h2"-1] =T2l2r+l(n) —±- • U (2r+V4p-1

Writing

(3.19) Q4p. , = (- If V 4 p * 12-2p+4(4p)! V - " J * % <2r+1)4p-1

(3.18) gives with the aid of (3.6) and (3.19) n

(3.20) £ (4n4p2) Q*P-1 = (4» + U(4n+2) { 1 + (~1)nH21'2n] .

This is the third of the recursive relations and may be compared directly in form with (3.13) and (3.16). &4P-i can also be expressed in terms of the Bernoulli numbers.

From (3.20) we deduce

±\»»r»>~\&-\t^&\\±&& n=1 {p=1 ) (k=0 )

or, after some manipulation,

(4p)! coshx- cos*

where as before 2a - 1 + i.

E n *4p - „ 2 1 cosh x + cos x - 2 cosh ax - 2 cos ax + 2\ 4p~1 WW " * TZ^Z—TZrz *

1976] THE SUMS OF CERTAIN SERIES COWTAIWIWG HYPERBOLIC FUNCTIONS 221

The right-hand side of (3.21) can be expressed as

•y ( coth y- coth l-~- -2 coth ax coth fax - 2 cosec ax cosech fax - tanh ^ tanh '^- i .

Recalling the expansions for coth x, cosech x already used and noting that in [1] for tanhx we obtain after some manipulation

nA , = (-IPfail V> / ,,<? (24P-2«- 1)(22«- 1) B g Q4P-1 ^.s 2 - , '-11 (2g)!(4p-2g)l B"B*">

and hence by (3.19)

(3 a , f ^(2r+1>l ^ 2T1 (24p-2q _ 1)(22q _ 1}

The expression in (3.11) can be differentiated as many times as we wish with respect t o * and y at points within the rectangle.

Differentiating once with respect to x and once with respect to y gives oo

(3.23) (2n)(4n - 1)i[(x + iy)4n~2 - (x - iy)4n"2] = Y* ^P^-1 /sinh ry sin rx + sinh rx sin ry] " sinh m r=1

Puttings = y = n/2 in (3.23) and defining /?4P„3 by

(3.24) R4p.3 = / - f " . fa ~ 2)1 E S ^ " T n4p-322p-1 ~0 (2r+ D4"-3 cosh ̂ +1)f

yields

(3.25) '^^--EUAK-* The quantities /?4P-j can be expressed in terms of the Euler numbers (see [1]).

Following a procedure similar to earlier ones, we can deduce from (3.25) that

Since

X>„ + , ^ r , -jr-ef "* 2 X4* lp+1 (4p+2U '

sec* oo

= E q=0

1 * 2"

^

-secM

x2« (2q)l '

where Elf E2, —, are the Euler numbers and E0 is taken as unity, we obtain

(3.26) R4p+1 - (4P ,2), £ £ E M,« ^ ^ 7 and hence

222 THE SUMS OF CERTAIN SERIES CONTAINING HYPERBOLIC FUNCTIONS [OCT.

00 2p n

/o97v V (-W „4p+U-4p~3 X^ (-1rEqE2p-q n - n 1 9 (3-27) S ^ / ^ W ^ / , ? * ' h <*»«!-»!>' p' °'l2' 2 Putting /? = / i n (3.23) yields

(3.28) xy = 2ft 2^ ™ ~ — | sinh ry sin rx + sinh rx sin ry I (~1)r

i

Hence differentiating once with respect t o * and then y we have, on putt ing* = y = n/2

~ tr+f

'-"ES; A / - / / sinh m

If we differentiate (3.28) (2p + 1) times with respect to x and (2p + 1) times with respect to y then f o r * -y = it/ 2 we find

± J ^ l l = O t P=1,2,.... *-^ smh nr r=/

Likewise differentiating (3.28) (2p) times with respect to x and 2p times with respect to y leads to

f " (2r+1)4p~1(-1)r

% cosb(2r+1)Z =0< P= 7<2<~"

We now proceed to find the sum of the last of the series referred to in the Introduction. Using the results of Section 2, it can be shown for n = 1, 2, •

- I (x + iy)4n+2 + (x- iy)4n+2 \ = (- 7)n<n4n22n(x2 -y2) + Y" (-1)r ^ o s h ^ cos ry - cosh ry cos rx ! 2 % > *-*> r sinh nr

r=1

o=1 r )

X J % i-i>"+1-P i d H ^ l 92n-2p+2,AnU (4n+2\\

P=1

The constant appearing in the Neumann solution is determined here to be zero by observing that each side of (3.29) vanishes when x = y = 0.

Putting* = 7r , / = 0 in (3.29) and de f in ing#4^ / by

(3.30) M4p+1 = (-1)p+1^4p"12'2p+2(4p)! £ / * M / * coshr^ p s , z . leads to "1 r p H sinh nr

(3.3D J2 M*p+i {4n4p2) = i + (-vnHr2n> P=1

From the recurrence relation (3.31) we deduce

\ T * M x p - 1 J. i ™+ a* +*n fax i + „ „ ax nn+ /'ax

2-r M*P+I J4$T ~ 1 2 T x T ~Jm Y T~ P=I

and hence

1976] THE SUMS OF CERTAIW SERIES CONTAINING HYPERBOLIC FUNCTIONS 223

22p-2 »++, - (-if -gg E h1)S (2mP2-2S~+2), B°B2^-°

Since 00 / -,r*/ u * °° ( coth^-2""4pcoth2ttr ) y ^ f-/A # cosh HT* / - -V1 } 2 f

~i r**'sinhnr ~ %1 } &+1 \

we can, noting (3.30), obtain the required sum. It also follows for/7 > 3 we can obtain a good approximation to

- cothJJ Za 4p+1

in terms of the Bernoulli numbers.

ACKNOWLEDGEMENTS

I am indebted to Professor L. Carlitz for pointing out to me how the result (3.17) can be deduced from (3,16). This result, I am told by Professor H. Halberstam, is contained, in a somewhat disguised form, in one of Ramanuj:.v s notebooks. I also thank Bruce Berndt for drawing my attention to the papers by Phillips and Sandham.

REFERENCES

1. R. 1 I'A. Bromwich, An Introduction to the Theory of Infinite Series, IVIacMillan & Co. Ltd., Second Edition, (1926), pp. 297-299.

2. Kantorovich and Krylov, Approximate Methods ofHigher Analysis (translated by Curtis D. Benster), P. Boordhoff Ltd., Gronigen, Netherlands (1958), pp. 7-9.

3. E. G. Phillips, "Note on Summation of Series," Journal London Math. Society, 4 (1929), pp. 114-116. 4. S. Ramanujan, Collected Papers, Edited by G. H. Hardy, et al, Cambridge University Press, 1927. 5. H. F. Sandham, "Some Infinite Series," Proc. Amer. Math. Soc, 5 (1954), pp. 430-436. 6. G. N. Watson, "Theorems Stated by Ramanujan (II)," J. London Math. Society, 3 (1928), pp. 216-225.

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